If $A$ is a subobject of $B$, and $B$ a subobject of $A$, are they isomorphic?

In category theory, a subobject of $X$ is defined as an object $Y$ with a monomorphism, from $Y$ to $X$. If $A$ is a subobject of $B$, and $B$ a subobject of $A$, are they isomorphic? It is not true in general that having monomorphisms going both ways between two objects is sufficient for isomorphy, so it would seem the answer is no.

I ask because I'm working through the exercises in Geroch's Mathematical Physics, and one of them asks you to prove that the relation "is a subobject of" is reflexive, transitive and antisymmetric. But it can't be antisymmetric if I'm right...

I don't think this can be true in general. What if we just take the category consisting of two objects $A$, $B$ and take morphisms $f:A\to B$, $g:B\to A$ with no relations between the morphisms, but forcing associativity. Then certainly $f$ and $g$ are monomorphisms but $A$ and $B$ are not isomorphic (since there are no relations between the morphisms).

Here is an excellent paper on that question for a bunch of different categories. It's true for any set-based category of "finite" things.

The paper (The Cantor–Schroeder–Bernstein property in categories by Don Laackman) defines a category $\mathcal{C}$ to have the CSB property to be if whenever $f : A \to B$ and $g : B \to A$ are monomorphisms in $\mathcal{C}$, then $A$ and $B$ are isomorphic. The categories of sets and well-ordered sets have this property, while the categories of topological spaces, groups, posets, or abelian groups don't. The first theorem of this paper is:

Theorem. If a category $\mathcal C$ has a faithful functor $F : \mathcal{C} \to \mathsf{FinSet}$ to the category of finite sets, then $\mathcal C$ has the CSB property.